What should be the least number of years in which the simple interest on Rs 2600 at `6(2)/(3)` % will be an exact number of rupees ?

Options

A. 3
B. 2
C. 5
D. 4

Answer

To solve the problem step by step, we need to determine the least number of years (T) for which the simple interest on Rs 2600 at a rate of \(6 \frac{2}{3}\%\) becomes an exact number of rupees.

### Step 1: Convert the percentage to a fraction
The interest rate given is \(6 \frac{2}{3}\%\). We can convert this mixed number into an improper fraction:
\[ 6 \frac{2}{3} = \frac{20}{3}\% \]

### Step 2: Write the formula for Simple Interest
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \(P\) = Principal amount (Rs 2600)
- \(R\) = Rate of interest (\(\frac{20}{3}\%\))
- \(T\) = Time in years (which we need to find)

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ SI = \frac{2600 \times \frac{20}{3} \times T}{100} \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ SI = \frac{2600 \times 20 \times T}{300} \]
\[ SI = \frac{52000 \times T}{300} \]
\[ SI = \frac{520T}{3} \]

### Step 5: Determine the condition for SI to be an exact number
For the Simple Interest (SI) to be an exact number of rupees, \(\frac{520T}{3}\) must be an integer. This means that \(520T\) must be divisible by 3.

### Step 6: Find the least value of T
Now, we need to find the least value of \(T\) such that \(520T\) is divisible by 3. We can check the divisibility of 520 by 3:
\[ 520 \div 3 = 173.33 \quad (\text{not divisible}) \]
The remainder when 520 is divided by 3 is 1 (since \(520 = 3 \times 173 + 1\)). Therefore, for \(520T\) to be divisible by 3, \(T\) must be such that:
\[ T \equiv 2 \mod 3 \]
The smallest positive integer satisfying this condition is \(T = 3\).

### Conclusion
Thus, the least number of years in which the simple interest on Rs 2600 at \(6 \frac{2}{3}\%\) will be an exact number of rupees is:
\[ \boxed{3} \]